where \(\tilde{\Gamma }_2^{ \mathrm {\scriptscriptstyle T} }\tilde{\Gamma }_2 = VW^{-1}V\)  and \(\tilde{\Gamma }^{ \mathrm {\scriptscriptstyle T} }_1\tilde{\Gamma }_1 = V\) . Such choices of \(\tilde{\Gamma }_1\)  and \(\tilde{\Gamma }_2\)  are not unique. It may be important that the mapping defined by \(\tilde{\Gamma }\)  preserve direction of asymmetries. For this reason, following [1]}, we use singular value decompositions to obtain \(\tilde{\Gamma }_1\)  and \(\tilde{\Gamma }_2\) .
